{
  "id": "2401.01041",
  "version": "v1",
  "published": "2023-10-10T11:41:17.000Z",
  "updated": "2023-10-10T11:41:17.000Z",
  "title": "Short proof of the Kneser-Edmonds theorem on the degree of a map between closed surfaces",
  "authors": [
    "Andrey Ryabichev"
  ],
  "comment": "3 pages with no figures (the proof takes less than 1 page)",
  "categories": [
    "math.GT"
  ],
  "abstract": "Suppose for closed surfaces $M,N$ there exists a continuous map $f:M\\to N$ of geometric degree $d>0$. Then $\\chi(M)\\le d\\cdot\\chi(N)$. This inequality was first proved by Kneser in case of orientable surfaces and by Edmonds for arbitrary $M,N$. We give a new simple proof of this result. Our proof is completely elementary and does not use additional techniques (such as the factorisation theorem of Edmonds and the absolute degree theory of Hopf).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T11:41:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed surfaces",
      "kneser-edmonds theorem",
      "short proof",
      "absolute degree theory",
      "simple proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}